The drop shadow is basically a transparent blurred black copy of the shape, so you can as well make it by yourself as a separate shape. That copy can have a mask (=clip or transparency, both work) You can also convert the shadow effect to a shape by expanding the appearance:
Warning: You have got a suggestion "use gradients". That's no-nonsense because expanding blurry effects generates raster images which cannot be scaled freely like vectors. See it in the Appearance panel.
Expanding drop shadow fortunately saves the original shape, but generates a group which has a raster image as the shadow. You must be careful you have high enough raster effect resolution in the document settings.
Unfortunately gradients are tricky for complex shapes, you may need a gradient mesh.
In Illustrator you keep the scalability if your shadow is a separate masked blurred shape or a masked copy of the whole shape which has the drop shadow effect. Scaling problems due rasterization start if you export to another vector formats. For example a shape with Gaussian Blur effect exported as SVG opens as a raster image in Inkscape.
Gradient meshes also cause export problems. Only the ordinary linear and radial gradients are well supported in other vector formats.